Solving linear program as linear system in polynomial time

Abstract

a b s t r a c tA physically concise polynomial-time iterative-cum-non-iterative algorithm is presented to solve the linear program (LP) Min c t x subject to Ax = b, x ≥ 0. The iterative part -a variation of Karmarkar projective transformation algorithm -is essentially due to Barnes only to the extent of detection of basic variables of the LP taking advantage of monotonic convergence. It involves much less number of iterations than those in the Karmarkar projective transformation algorithm. The shrunk linear system containing only the basic variables of the solution vector x resulting from Ax = b is then solved in the mathematically non-iterative part. The solution is then tested for optimality and is usually more accurate because of reduced computation and has less computational and storage complexity due to smaller order of the system. The computational complexity of the combination of these two parts of the algorithm is polynomial-time O(n 3 ). The boundedness of the solution, multiple solutions, and no-solution (inconsistency) cases are discussed. The effect of degeneracy of the primal linear program and/or its dual on the uniqueness of the optimal solution is mentioned. The algorithm including optimality test is implemented in Matlab which is found to be useful for solving many real-world problems.

Energy‐efficient task scheduling on heterogeneous computing systems by linear programming

“…Consequently, an energy-efficient schedule can be provided in a straightforward manner by a linear programming solver that has a demonstrated ability to offer an optimal solution in polynomial time. [32][33][34] The main contributions of our work are as follows.…”

“…Such a combination is able to fully utilize the slack time for a particular task, thus achieving the local optimality of energy savings. 24,25 Another advantage of using nonbinary duty factors is that a linear program with real variables can be solved in polynomial time, [32][33][34] whereas binary programming has been shown to be NP-complete. 38 Furthermore, since the computational difficulties in the solution of a linear program are caused to a considerable degree by their dimensions, a reduction in the number of variables is essential for a reasonable expenditure.…”

“…Such a combination is able to fully utilize the slack time for a particular task, thus achieving the local optimality of energy savings . Another advantage of using nonbinary duty factors is that a linear program with real variables can be solved in polynomial time, whereas binary programming has been shown to be NP‐complete …”

“…Then, the goal of energy reduction is formulated as a linear objective function, while the precedence and deadline constraints implied in the given schedule are also represented in a linear format. Consequently, an energy‐efficient schedule can be provided in a straightforward manner by a linear programming solver that has a demonstrated ability to offer an optimal solution in polynomial time . The main contributions of our work are as follows.…”